While it would appear that it is primarily its connectedness that
prevents the familiar double cover of the circle from admitting a global
section, the impression that one must have connectedness for such examples
to be available is illusory. To illustrate, we exhibit a double cover of
Heath’s (very disconnected) V-space likewise admitting no global section.
The motivation for finding such a disconnected example came from topos
theory; we will present both the example and that motivation.